Final answer:
The distance between the two parallel chords in a circle of radius 13 cm is found using the Pythagorean theorem. The heights from the center to each chord are calculated for right-angled triangles formed with the radius and the distances are added to find the total separation, resulting in 17 cm.
Step-by-step explanation:
To find the distance between the two chords, we will use the properties of right-angled triangles within the circle. Since the chords are on opposite sides of the center, we have two triangles with a common height from the center to the midpoint of each chord (perpendicular from the center to a chord bisects the chord), and the hypotenuse being the radius of the circle.
Let's consider the larger chord first. The hypotenuse is the radius of the circle, 13 cm, and the half of the chord (which will be the base of the right-angled triangle) is half of 24 cm, which is 12 cm. We can use the Pythagorean theorem to solve for the height (h1) from the center to the chord.
13^2 = h1^2 + 12²
169 = h1^2 + 144
h1^2 = 169 - 144
h1^2 = 25
h1 = 5 cm
For the smaller chord, again, the hypotenuse is the radius of the circle, 13 cm, and the half of the chord is half of 10 cm, which is 5 cm. Applying the Pythagorean theorem once more to solve for the height (h2) from the center to this chord, we have:
13^2 = h2² + 5^2
169 = h2^2 + 25
h2^2 = 169 - 25
h2^2 = 144
h2 = 12 cm
The distance between the two chords will be the sum of the heights h1 and h2: 5 cm + 12 cm = 17 cm. However, the options provided do not contain this value, which suggests there might be an error in the statement of the question or the options provided.